Advertisement

Introduction

  • D. M. Gitman
  • I. V. Tyutin
  • B. L. Voronov
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 62)

Abstract

We call attention to the fact that quantization includes the problem of a correct definition of quantum-mechanical observables like Hamiltonian, momentum, etc. as self-adjoint operators in an appropriate Hilbert space. This problem is nontrivial for systems on nontrivial manifolds or/and with singular interactions. A naïve treatment based on the experience in finite-dimensional algebra or even infinite-dimensional algebra with bounded operators can result in paradoxes and incorrect results. A set of such paradoxes is considered.

Keywords

Hilbert Space Quantum Mechanic Momentum Operator Symmetric Operator Unbounded Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 23.
    le Bellac, M.: Quantum Physics. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  2. 25.
    Berezin, F.A.: The Method of Second Quantization. Academic Press, New York (1966)MATHGoogle Scholar
  3. 26.
    Berezin, F.A., Faddeev, L.D.: A remark on Schrödinger’s equation with a singular potential. Sov. Math. Dokl. 2, 372–375 (1961)MATHGoogle Scholar
  4. 27.
    Berezin, F.A., Shubin, M.A.: Schrödinger Equation. Kluwer, New York (1991)MATHCrossRefGoogle Scholar
  5. 31.
    Bonneau, G., Faraut, J., Valent, G.: Self-adjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322–331 (2001)CrossRefGoogle Scholar
  6. 32.
    Bohm, D.: Quantum Theory. Prentice-Hall, Englewood Cliffs, NJ (1951)Google Scholar
  7. 37.
    Capri, A.: Nonrelativistic Quantum Mechanics. World Scientific Publishers, Singapore (2002)Google Scholar
  8. 39.
    Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)Google Scholar
  9. 44.
    Davydov, A.S.: Quantum Mechanics, 2nd edn. Pergamon Press, Oxford/New York (1976)Google Scholar
  10. 48.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)MATHGoogle Scholar
  11. 49.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York (1964)Google Scholar
  12. 57.
    Faddeev, L.D., Yakubovsky, O.A.: Lectures on Qunatum Mechanics. Leningrad State University Press, Leningrad (1980)Google Scholar
  13. 63.
    Galindo, A., Pascual, P.: Quantum Mechanics, vols. 1 and 2. Springer (1990, 1991)Google Scholar
  14. 64.
    Gasiorowicz, S.: Quantum Physics. Wiley, New York (1974)Google Scholar
  15. 74.
    Gieres, F.: Mathematical surprises and Dirac’s formalism in quantum mechanics. Rep. Prog. Phys. 63, 1893–1931 (2000)CrossRefGoogle Scholar
  16. 75.
    Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints. Springer, Berlin (1990)MATHCrossRefGoogle Scholar
  17. 83.
    Gustafson, S.J., Sigal, I.M.: Mathematical Concepts of Quantum Mechanics. Universitext. Springer, Berlin (2003)MATHCrossRefGoogle Scholar
  18. 84.
    Haag, R.: Local Quantum Physics. Springer, Berlin (1996)MATHGoogle Scholar
  19. 91.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)MATHGoogle Scholar
  20. 98.
    Konishi, K., Paffuti, G.: Quantum Mechanics: A New Introduction. Oxford University Press, Oxford (2009)MATHGoogle Scholar
  21. 104.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford (1977)Google Scholar
  22. 109.
    Liboff, R.L.: Introduction to Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  23. 112.
    Messiah, A.: Quantum Mechanics. Interscience, New York (1961)Google Scholar
  24. 128.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Functional Analysis. Academic Press, New York (1980)Google Scholar
  25. 136.
    Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  26. 138.
    Schiff, L.I.: Quantum Mechanics. McGraw–Hill, New York (1955)Google Scholar
  27. 144.
    Takhtajan, L.A.: Quantum Mechanics for Mathematicians. Graduate Studies in Mathematics, 95. American Mathematical Society (2008)Google Scholar
  28. 147.
    Thirring, W.: Quantum Mathematical Physics – Atoms, Molecules and Large Systems. Springer, Berlin (2002)Google Scholar
  29. 153.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)MATHGoogle Scholar
  30. 156.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators.I. Russ. Phys. J. 50(1) 1–31 (2007)Google Scholar
  31. 157.
    B.L. Voronov, D.M. Gitman, I.V.Tyutin, Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators, Russ. Phys. J. 50/9 853–884 (2007)Google Scholar
  32. 158.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions. Russ. Phys. J. 51(2) 115–157 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • D. M. Gitman
    • 1
  • I. V. Tyutin
    • 2
  • B. L. Voronov
    • 2
  1. 1.Instituto de FísicaUniversidade de São PauloSão PauloBrasil
  2. 2.Department of Theoretical PhysicsP.N. Lebedev Physical InstituteMoscowRussia

Personalised recommendations